Question

Tính giá trị biểu thức :

                  \(P=\sqrt[3]{6+\sqrt{\frac{847}{27}}}+\sqrt[3]{6-\sqrt{\frac{847}{27}}}\)

Answer 1

\(P=\sqrt[3]{6+\sqrt{\frac{847}{27}}}+\sqrt[3]{6+\sqrt{\frac{847}{27}}}\)

Ta áp dụng hằng đẳng thức : 

\(\left(a+b\right)^3=a^3+b^3+3ab\left(a+b\right)\)

\(\Rightarrow P^3=6+\sqrt{\frac{847}{27}}+6-\sqrt{\frac{847}{27}}+3\sqrt[3]{6+\sqrt{\frac{847}{27}}}.\sqrt[3]{6-\sqrt{\frac{847}{27}}}\left(3\sqrt[3]{6+\sqrt{\frac{847}{27}}}.\sqrt[3]{6-\sqrt{\frac{847}{27}}}\right)\)

\(\Leftrightarrow P^3=12+3.\sqrt[3]{36-\frac{847}{27}}.P=12+5P\)

\(\Leftrightarrow P^3-5P-12=0\)

\(\Leftrightarrow\left(P-3\right)\left(P^2+3P+4\right)=0\)

\(\Leftrightarrow P=3\) hoặc \(P^3+3P+4=0\) vô nghiệm

Vậy \(P=3\)